Li1.3Al0.3Ti1.7(PO4)(3) (LATP) materials are made of a three-dimensional framework of TiO6 octahedra and PO4 tetrahedra, which provides several positions for Li+ ions. The resulting high ionic conductivity is promising to yield electrolytes for all-solid-state Li-ion batteries. In order to elaborate dense ceramics, conventional sintering methods often use high temperature (>= 1000 degrees C) with long dwelling times (several hours) to achieve high relative density (similar to 90%). In this work, an innovative synthesis and processing approach is proposed. A fast and easy processing technique called microwave-assisted reactive sintering is used to both synthesize and sinter LATP ceramics with suitable properties in one single step. Pure and crystalline LATP ceramics can be achieved in only 10 min at 890 degrees C starting from amorphous, compacted LATP's precursors powders. Despite a relative density of 88%, the ionic conductivity measured at ambient temperature (3.15 x 10(-4) S cm(-1)) is among the best reported so far. The study of the activation energy for Li+ conduction confirms the high quality of the ceramic (purity and crystallinity) achieved by using this new approach, thus emphasizing its interest for making ion conducting ceramics in a simple and fast way.